Maximum distance separable (MDS) array codes constitute an important class of error-correcting codes due to their optimal distance properties and their relevance in distributed storage systems. In this paper, we investigate the construction and decoding of MDS array codes over $\mathbb{F}_q^b$ based on superregular matrices, with emphasis on superregular Vandermonde and Cauchy matrices. We propose decoding algorithms for [n,k,d] MDS array codes, where n=m+k and d=m+1, capable of correcting symbol errors without prior knowledge of their locations. Unlike existing approaches restricted to specific parameter settings, the proposed algorithms apply to general configurations and rely on algebraic relations that do not follow from straightforward extensions of previous methods. Specifically, these algorithms correct one symbol error for $m \geq 2$ and two symbol errors for $m \geq 4$. For the two-error case, the decoding procedure admits a simplified form when Vandermonde superregular matrices are employed, reducing computational complexity. We analyze the algebraic structure of the three-symbol-error case, focusing on the most involved configuration in which all errors occur in information symbols, and we discuss how the method may be extended to the general case. These algorithms are computationally efficient for moderate parameter sizes, as they rely on structured algebraic operations over $\mathbb{F}_q^b$ and the solution of small linear systems, making them suitable for distributed storage applications. From an application perspective, the proposed approach provides a flexible alternative to RAID~6 schemes. Unlike RAID~6, which is limited to two parity disks and often requires prior knowledge of error locations, our construction supports general configurations and enables the correction of multiple symbol errors without location information, at the cost of increased algebraic complexity

翻译：最大距离可分（MDS）阵列码因其最优距离特性及在分布式存储系统中的重要应用，构成了一类重要的纠错码。本文研究基于超正则矩阵（重点分析超正则范德蒙德矩阵和柯西矩阵）的$\mathbb{F}_q^b$上MDS阵列码的构造与译码问题。针对$[n,k,d]$ MDS阵列码（其中$n=m+k$，$d=m+1$），我们提出无需先验错误位置信息的符号纠错译码算法。不同于现有方法局限于特定参数设置，本文算法适用于一般配置，且依赖非前方法直接扩展所得的代数关系。具体而言，该算法可实现$m \geq 2$时的单符号纠错与$m \geq 4$时的双符号纠错。在双符号纠错场景中，当采用范德蒙德超正则矩阵时，译码过程可简化为简化形式，从而降低计算复杂度。针对三符号纠错情况，我们重点分析所有错误均发生于信息符号这一最复杂配置的代数结构，并讨论将该方法推广至一般情形的途径。由于算法依赖于$\mathbb{F}_q^b$上的结构化代数运算及小型线性系统求解，在中等参数规模下具有计算高效性，适用于分布式存储应用。从应用视角看，本文方法为RAID~6方案提供了灵活替代方案——RAID~6仅支持双校验盘且常需错误位置先验信息，而本文构造支持通用配置，无需位置信息即可纠正多符号错误，其代价是增加了代数复杂度。